New Finslerian Ressiner-Nordstrom spacetime with constant flag curvature.
problem Exploring a new spacetime model with constant flag curvature.
method Derived Finslerian Ressiner-Nordstrom solution and analyzed its properties.
result The solution differs from Ressiner-Nordstrom metric only in two dimensional subspace with constant flag curvature.
The notion of wind Finslerian structure is developed; this is a generalization of Finsler metrics where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which …
The Sagnac effect is re-examined using Finslerian geometry.
problem Understanding the Sagnac effect in general relativity.
method Reviewing the geometry of the Sagnac effect with a focus on Finslerian metrics.
result An asymmetry in Finslerian metrics affects the Sagnac effect for both future-pointing null and timelike geodesics.
Criteria for completeness and existence of Cauchy hypersurfaces in spacetimes.
problem Characterizing completeness and existence of Cauchy hypersurfaces in spacetimes.
method Introducing wind Riemannian structures and deriving criteria for completeness and existence of Cauchy hypersurfaces.
result Simple criteria for slices of spacetimes to be Cauchy.
New findings on lightconvex boundaries in Finslerian spacetimes.
problem Understanding causal structures in Finslerian spacetimes.
method General results for indefinite Finslerian manifolds with boundary, and equivalence among boundary lightconvexity, causally simple interior, and Hausdorff space of cone geodesics.
result Equivalence among boundary lightconvexity, causally simple interior, and Hausdorff space of cone geodesics in globally hyperbolic spacetimes.
We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinat…
We introduce the notion of a standard static Finsler spacetime where the base is a Finsler manifold. We prove some results which connect causality with the Finslerian geometry of the base extending analogous ones for static and stationary Lorentzian spacetimes.
We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.
Study extends Huygens' principle to Finsler spaces, with applications in gravity.
problem Validating Huygens' principle in Finsler spaces for wavefronts.
method Extending a theorem to n-dimensional Finsler spaces and applying to analogue gravity.
result Finsler geometry reveals directional spacetime structures like horizons and ergospheres.
The paper explores Finsler-type objects and their variational problems on spacetimes.
problem Generalizing Einstein equations to the Finsler setting.
method Study of the ladder of Finsler-type objects and their variational problems.
result Application of the ladder structure to variational proposals for Finsler spacetimes.
The paper proposes a notion of volume element for Finsler spaces with metrics of Lorentzian signature, equipped with a time orientation. This notion is based on a slight modification of the idea of Holmes-Thompson volume element working for positive definite Finsler metrics and can be used in field-theoretical applicat…
Study on Finsler spacetimes with timelike Killing vectors, introducing stationary splitting.
problem Characterizing Finsler spacetimes with timelike Killing vectors.
method Introducing a new class of stationary splitting Finsler spacetimes and characterizing their properties.
result A Finsler spacetime with a timelike Killing vector field is locally a stationary splitting.
The paper proposes extensions of the usual notions of Finslerian volume to time orientable Finsler spacetime manifolds. The basic idea is to replace, in the classical Busemann-Hausdorff and Holmes-Thompson definitions, integration on the indicatrices of the given metric (which are, in Lorentzian signature, non-compact,…
Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary ∂V for a standard conformally stationary spacetime V = R x M, suggests a natural compactification MB associated to any Riemannian metric on M or, more generally, to any Fin…
Study Berwald spacetimes and their relation to very special relativity.
problem Understanding vacuum dynamics in Berwald spacetimes.
method Derived Finsler generalization of Einstein's equations and derived conditions for Berwald spacetimes.
result Identified necessary and sufficient conditions for VGR line elements to be Berwald type.
In this paper we argue that when gauge invariance is taken into consideration, there is no consistent geometric framework of Finsler class that can accommodate Randers type spaces. In this context, an alternative non-Finslerian framework for Randers spacetimes compatible with gauge invariance is introduced.
Unified geometry for relativity and beyond.
problem Unified geometric framework for relativity and beyond.
method Unified Lorentz-Finsler geometry.
result Unified framework for relativity and beyond.
Study examines causal properties of Finsler spacetimes with cone Killing vectors.
problem Characterize causality in Finsler spacetimes with specific Killing vectors.
method Explores the relationship between wind Riemannian structures and spacetimes with cone Killing vectors, focusing on Finsler-Kropina metrics.
result Characterizes causality properties using metric-type properties of Finslerian structures.
We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullst…
Study of conformal mappings in pseudo-Finsler spaces, extending results from pseudo-Riemannian geometry.
problem Understanding conformal mappings in pseudo-Finsler spaces and their properties.
method Extending results from pseudo-Riemannian geometry and proposing a technique to reduce problems involving pseudo-Finslerian conformal vector fields.
result Conformal groups of flat pseudo-Finsler spaces can be richer than flat Finslerian and pseudo-Euclidean conformal groups.
Establishes conditions for Berwald Finsler geometries.
problem Identifying Berwald Finsler geometries among Finsler spaces.
method Develops a first order partial differential equation for Berwald Finsler Lagrangians.
result Generalizes earlier findings on Berwald conditions for specific geometries.
We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on M=R×S and Randers metrics on S. In particular, for stationary spacetimes, we give a simple characterization of when they are causally conti…
Introduces Finslerian convolution metrics and their properties.
problem No specific problem stated; focuses on new metric concept.
method Definition and study of Finslerian convolution metrics.
result Characterization of Finslerian convolution metrics of Riemannian, Minkowskian, and Randers types.
Wind Riemannian structures generalize Randers metrics and are classified for constant flag curvature.
problem Classifying wind Riemannian structures of constant flag curvature.
method Using the natural data for Zermelo navigation problem, constructing WRS from a Riemannian metric and a vector field (wind).
result Local and global classification of wind Riemannian structures of constant flag curvature.
Anisotropic connections and parallel transport defined in Finsler spacetimes.
problem Defining and characterizing anisotropic connections and parallel transport in Finsler spacetimes.
method Introducing a new covariant derivative and parallel transport, identifying vertically trivial Finsler connections with anisotropic connections, and characterizing the Levi-Civita-Chern anisotropic connection.
result Characterization of the Levi-Civita-Chern anisotropic connection as the one preserving the length of parallely propagated vectors.
The paper proves inequalities for optimal transport on sub-Finslerian manifolds.
problem Optimal transport inequalities on sub-Finslerian manifolds.
method Introduction of sub-Finslerian Jacobi fields and optimal transport theory.
result Characterization of generalized distortion coefficients and fundamental geometric inequalities.
Lecture notes on Finslerian geometry.
problem No specific problem stated; covers Finslerian geometry.
method Lecture notes.
result No specific key result mentioned.
In this work, convergence of evolving Finslerian metrics first in a general flow next under Finslerian Ricci flow is studied. More intuitively it is proved that a family of Finslerian metrics g(t) which are solutions to the Finslerian Ricci flow converge in C∞ to a smooth limit Finslerian metric as t ap…
New proof of Finslerian Beltrami theorem in 2D.
problem Generalizing Beltrami's theorem to Finslerian geometry.
method Presented a new proof of the theorem.
result Validated the theorem in Finslerian geometry, including 2D.
Classifies Finslerian metrics locally conformally R-Einstein.
problem Characterizing Finslerian metrics with specific curvature properties.
method Using the pulled-back tangent bundle approach to Finslerian Geometry, an intrinsic characterization of R-Einstein metrics is given, leading to the classification of locally conformally R-Einstein metrics.
result Finslerian metrics locally conformally R-Einstein are classified.
Revisits stress-energy tensor in Finsler spacetimes, showing it's anisotropic.
problem Defining stress-energy tensor in Finsler spacetimes.
method Uses both heuristic and Lagrangian approaches, revisits divergence and conservation laws.
result Introduces a natural anisotropic Lie bracket derivation leading to the Chern anisotropic connection.
Classification of Finslerian spaces with nontrivial concircular transformations.
problem Classifying Finslerian spaces with nontrivial concircular transformations.
method Proving the existence of at most two critical points in a conformal circle-preserving transformation and presenting a diffeomorphism classification based on these critical points.
result Presented a diffeomorphism classification of Finslerian manifolds that admit nontrivial conformal circle-preserving transformations.
The Hopf-Rinow theorem is extended to sub-Finslerian geometry.
problem Extending the Hopf-Rinow theorem to sub-Finslerian manifolds.
method Investigation of sub-Finslerian bundle, exponential map, and Legendre transformation.
result Established a relation between completeness, geodesic completeness, and compactness in sub-Finslerian geometry.
Study on generalized m-Kropina metrics in modified gravity and cosmology.
problem Understanding the geometric properties and applications of generalized m-Kropina metrics. method Proving the rationality of Finslerian geometric objects and studying the conditions for Einstein metrics.
result Conditions for a generalized m-Kropina metric to be an exact solution in modified gravity and cosmology. Researchers prove constant solutions for a specific Finslerian equation.
problem Investigating exponentially harmonic functions on Finslerian spaces.
method Analyzing the exponential energy functional and using nonnegative Ricci curvature conditions.
result Any bounded solution to the Finslerian equation is constant.
Finsler gravity vacuum equation reduces to Ricci vanishing under specific conditions.
problem Solving Einstein vacuum equations in Finsler gravity
method Identifying conditions for vacuum equation reduction
result Scalar Finsler gravity vacuum equation reduces to Ricci vanishing
Natural volume forms defined for pseudo-Finslerian manifolds with specific metrics.
problem Defining natural volume forms on pseudo-Finslerian manifolds with m-th root metrics. method Definitions depend on the parity of m, expressed in terms of Cayley hyperdeterminants. result Volume forms computation simplified by avoiding integration over the indicatrix.
Compact Finslerian manifolds don't admit non-trivial circle-preserving transformations.
problem Characterizing Finslerian manifolds without circle-preserving transformations.
method Analyzing critical points of conformal transformations to prove manifold rigidity.
result Compact Finslerian manifolds are Riemannian and conformally diffeomorphic to standard spheres, Euclidean spaces, or hyperbolic spaces.
This paper considers fundamental issues related to Finslerian isometries, submetries, distance and geodesics. It is shown that at each point of a Finsler manifold there is a distance coordinate system. Using distance coordinates, a simple proof is given for the Finslerian version of the Myers-Steenrod theorem and for t…
Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
Researchers estimate gradients of solutions to a Finslerian Allen-Cahn equation.
problem Estimating gradients of solutions to a specific type of partial differential equation.
method Using the Finslerian Allen-Cahn equation as an Euler-Lagrange equation to a Liapunov entropy functional, proving gradient estimates on compact and noncompact Finsler metric measure spaces.
result Global and local gradient estimates of positive solutions to the Finslerian Allen-Cahn equation.
The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class Ω…
Study on nonholonomic mechanics and sub-Finsler geometry.
problem Understanding nonholonomic mechanical systems and their geometric properties.
method Variational approach, sub-Finsler manifolds, nonholonomic sub-Finslerian structure.
result Existence and properties of extremals in nonholonomic mechanics.
Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.
problem Derives gradient estimate for a nonlinear parabolic equation on Finsler manifolds.
method Leverages a new Laplacian comparison theorem to derive a Li-Yau type gradient estimate.
result Establishes a Li-Yau type gradient estimate for the Finslerian logarithmic Schrödinger equation.
Paper develops methods for estimating gradients of Finslerian Schrödinger equations.
problem Estimating gradients of Finslerian Schrödinger equations.
method Develops new Laplacian comparison theorem and applies it to Finslerian Schrödinger equation.
result Global and local Li-Yau type gradient estimates for positive solutions.
Researchers propose and solve a class of pseudo-Finslerian metrics with angle-separation.
problem Characterizing and solving pseudo-Finslerian metrics with specific angle-separation conditions.
method Derived complete algebraic and differential equations, solved the set for angle-regular solutions.
result Found and described an angle-regular solution for the Finsleroid-in-pseudo-Finsleroid type.
The paper examines parallel one forms on Riemannian and Finslerian manifolds.
problem Existence of parallel one forms on Riemannian and Finslerian manifolds.
method Using Finslerian settings, the paper investigates the existence of parallel one forms on Riemannian manifolds and Finslerian manifolds, proving conditions for their existence and non-existence.
result Conditions for the existence and non-existence of parallel one forms on Riemannian and Finslerian manifolds.
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry